Theorem crnggrpd 20180

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20179	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20175	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2099  Grpcgrp 18883  CRingccrg 20167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-ring 20168  df-cring 20169

This theorem is referenced by:  fermltlchr  21452  psdmul  22083  psd1  22084  ply1chr  22218  ply1fermltlchr  22224  erlbr2d  32972  rlocaddval  32976  rloccring  32978  rloc0g  32979  rlocf1  32981  fracerl  32986  irngss  33355  irredminply  33378  algextdeglem4  33382  algextdeglem5  33383  aks6d1c1p3  41575  aks6d1c2lem4  41592  aks6d1c6lem2  41637  evlsbagval  41793  selvvvval  41812  evlselv  41814  selvadd  41815  prjcrv0  42051